Covariant Solutions of the Bethe-Salpeter Equation

Abstract

A fully covariant investigation is made of the BetheSalpeter equation for a pair of nucleons, with pseudoscalar interaction. The ladder'' approximation is adopted, but pair creation and nucleonic recoil are accoumted for exactly. Matrix solutions are obtained, with varying degrees of explicitness, for instantaneous and delayed interaction, vanishing and non- vanishing meson mass, and for vanishing and non-vanishing total energy. lmportant properties are disclosed which are either obscured or do not appear at all in nonrelativistic approximation. First the Bethe-Salpeter equation is reduced to a pair of coupled differential equations in which the Dirac matrices appear only in the coupling. In the instantaneous interaction approximation, these can be reduced to a single covariant equation in a single variable showing the radical influence of nucleon recoil on pair effects. When the instantaneous interaction approximation is discarded, new features appear. There is a discrete infinity of stable states corresponding to each one of the non-relativistic theories, requiring a new quantum number for their enumeration. Jastrow's hypothesis of a repulsive core'' interaction is rigorously established, and the singularity is isolated. It is shown how to obtain solutions corresponding to states of higher angular momentum from those with J = 0, making use ofmore » the relativistic quantum enumeration. The conclusion is drawn that the relativistic quantam number is a property of the state of any pair of interacting particles, and its possible connection with the strangeness'' number is discussed. (auth)« less